document.write( "Question 1149313: In a class of 36 students 25 study chemistry and 22 study mathematics and 25
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Algebra.Com's Answer #770703 by AnlytcPhil(1806)\"\" \"About 
You can put this solution on YOUR website!
We can solve this, but it contains a very badly-worded ambiguous statement.
>>15 study one of the three subjects<<
The word \"one\" could either mean \"EXACTLY one\" or \"ONE or more\".
>>15 study EXACTLY one of the three subjects<<
could give one answer and
>>15 study one OR MORE of the three subjects<<
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document.write( "could give another.  However, luckily we can tell which it is in this problem.\r\n" );
document.write( "We can tell that it means \"EXACTLY one\" because 15 happens to be too small for\r\n" );
document.write( "\"ONE or more\" to make sense, because, for instance, there are more than 15 in,\r\n" );
document.write( "say, the 25 that study chemistry.  So luckily we can solve the problem, but you\r\n" );
document.write( "should point out to your teacher not to just write \"one\" ever, but always\r\n" );
document.write( "\"EXACTLY one\" or \"ONE or more\".  For if you have $5, you have $1, right?\r\n" );
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document.write( "Anyway:\r\n" );
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\n" ); document.write( "In a class of 36 students 25 study chemistry and 22 study mathematics and 25
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document.write( "d+e+f+g+h+i+j+k = 36\r\n" );
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25 study chemistry
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document.write( "d+e+g+h = 25\r\n" );
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and 22 study mathematics
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document.write( "e+f+h+i = 22\r\n" );
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and 25 study physics
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document.write( "g+h+i+j = 25\r\n" );
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17 study physics and mathematics
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document.write( "h+i = 17\r\n" );
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18 study physics and chemistry
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document.write( "g+h = 18\r\n" );
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15 study EXACTLY one of the three subjects
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document.write( "d+f+j = 15\r\n" );
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document.write( "So we line up the equations:\r\n" );
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document.write( "1   d+e+f+g+h+i+j+k = 36\r\n" );
document.write( "2   d+e  +g+h       = 25\r\n" );
document.write( "3     e+f  +h+i     = 22\r\n" );
document.write( "4         g+h+i+j   = 25\r\n" );
document.write( "5           h+i     = 17\r\n" );
document.write( "6         g+h       = 18\r\n" );
document.write( "7   d  +f      +j   = 15\r\n" );
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document.write( "This is an \"under-determined system\" for it has 8 unknowns\r\n" );
document.write( "but only 7 equations.  The determining thing will be the fact\r\n" );
document.write( "that the solutions must all be non-negative integers.\r\n" );
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document.write( "We use matrix feature of TI-83 or 84.  And we get the\r\n" );
document.write( "answers in terms of k, the number studying none of the three \r\n" );
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document.write( "d = 6-k\r\n" );
document.write( "e = 1+k\r\n" );
document.write( "f = 4-k\r\n" );
document.write( "g = 3-2k\r\n" );
document.write( "h = 15+2k\r\n" );
document.write( "i = 2-2k\r\n" );
document.write( "j = 5+2k\r\n" );
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document.write( "Since i = 2-2k ≥ 0\r\n" );
document.write( "           -2k ≥ -2  \r\n" );
document.write( "             k ≤ 1\r\n" );
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document.write( "So there are two solutions, k=0 and k=1\r\n" );
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document.write( "Substituting k = 1 in the equations above gives\r\n" );
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document.write( "Substituting k = 0 in the equations above gives\r\n" );
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document.write( "Find a)Those who study all three subject \r\n" );
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document.write( "That's h, which is 15 for k=0, or 17 for k=1\r\n" );
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document.write( "b) only mathematics and chemistry \r\n" );
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document.write( "That's g, which is 3 for k=0, or 1 for k=1. \r\n" );
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document.write( "The second solution above with k=0 agrees with the other tutors.  But I do\r\n" );
document.write( "consider the first case k=1 to be a legitimate solution as well.\r\n" );
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document.write( "Edwin
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